The Original Problem in 2D

This is a 2-dimensional representation of the iteration rule . The points plotted are of the form . Drag point A and observe the motion of the individual points before you turn on the traces of the points. It is best to click on the point and use the arrow keys on your keyboard to move the point.

Can you explain why the points move along the paths they do?

Is it possible to find an equation that describes their paths?

What are some invariants?

Are there any characteristics of the traces that seem to persist regardless of the placement of point A?